Abstract

In this paper, we establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. It is worth pointing out that we remove some quite restrictive conditions in the corresponding results. An appropriate example, such that all conditions of this result are satisfied and that other conditions are not satisfied, is provided.

MSC:47H05, 47H09, 47H10.

Keywords

1 Introduction

Let X be a real Banach space with the dual space X∗. The value of f∈X∗ at x∈X is denoted by 〈x,f〉. The normal duality mapping J from X into a family of nonempty (by the Hahn-Banach theorem) weak-star compact subsets of X∗ is defined by

J(x)={f∈X∗:〈x,f〉=∥x∥2=∥f∥2},∀x∈X.

Let U={x∈X:∥x∥=1}. A Banach space X is said to be uniformly convex if for each ϵ∈(0,2], there exists δ>0 such that for any x,y∈U,

∥x−y∥≥ϵimplies∥x+y2∥≤1−δ.

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit

limt→0∥x+ty∥−∥x∥t

(1.1)

exists for all x,y∈U. It is said to be uniformly smooth if limit (1.1) is attained uniformly for x,y∈U. It is well known that if X is smooth, then J is single-valued and continuous from the norm topology of X to the weak-star topology of X∗, i.e., norm to weak∗ continuous. It is also well known that if X is uniformly smooth, then J is uniformly continuous on bounded subsets of X from the strong topology of X to the strong topology of X∗, i.e., uniformly norm-to-norm continuous on any bounded subset of X; see [1, 2] for more details.

Also, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:

ρ(τ)=sup{12(∥x+y∥+∥x−y∥)−1:x,y∈X,∥x∥=1,∥y∥=τ}.

It is known that X is uniformly smooth if and only if limτ→0ρ(τ)/τ=0. Let q be a fixed real number with 1<q≤2. Then a Banach space X is said to be q-uniformly smooth if there exists a constant c>0 such that ρ(τ)≤cτq for all τ>0. One should note that no Banach space is q-uniformly smooth for q>2; see [3] for more details. So, in this paper, we focus on a 2-uniformly smooth Banach space. It is well known that Hilbert spaces and Lebesgue Lp (p≥2) spaces are uniformly convex and 2-uniformly smooth.

Recall that a mapping T:X→X is said to be nonexpansive if

∥Tx−Ty∥≤∥x−y∥,∀x,y∈X.

A point x∈X is a fixed point of T if Tx=x. Let Fix(T) denote the set of fixed points of T; that is, Fix(T)={x∈X:Tx=x}.

A mapping f¯:X→X is called strongly pseudo-contractive if there exists a constant ρ∈(0,1) and j(x−y)∈J(x−y) satisfying

〈f¯(x)−f¯(y),j(x−y)〉≤ρ∥x−y∥2,∀x,y∈X.

A mapping f:X→X is a contraction if there exists a constant α∈(0,1) such that

∥f(x)−f(y)∥≤α∥x−y∥,∀x,y∈X.

Since 〈f(x)−f(y),j(x−y)〉≤∥f(x)−f(y)∥∥x−y∥≤α∥x−y∥2, we have that f is a strong pseudo-contraction.

Let η>0, a mapping F¯ of X into X is said to be η-strongly accretive if there exists j(x−y)∈J(x−y) such that

〈F¯x−F¯y,j(x−y)〉≥η∥x−y∥2

for all x,y∈X. A mapping F¯ of X into X is said to be k-Lipschitzian if, for k>0,

∥F¯x−F¯y∥≤k∥x−y∥

for all x,y∈X. It is well known that if X is a Hilbert space, then an η-strongly accretive operator coincides with an η-strongly monotone operator.

Yamada [4] introduced the following hybrid iterative method for solving the variational inequality in a Hilbert space:

xn+1=Txn−μλnF(Txn),n≥0,

(1.2)

where F is a k-Lipschitzian and η-strongly monotone operator with k>0, η>0 and 0<μ<2η/k2. Let a sequence {λn} of real numbers in (0,1) satisfy the conditions below:

(A1)limn→∞λn=0,(A2)∑n=0∞λn=∞,(A3)limn→∞(λn−λn+1)/λn+12=0.

He proved that {xn} generated by (1.2) converges strongly to the unique solution of the variational inequality

〈Fx˜,x−x˜〉≥0,∀x∈Fix(T).

An example of sequence {λn} which satisfies conditions (A1)-(A3) is given by λn=1/nσ, where 0<σ<1. We note that condition (A3) was first used by Lions [5]. It was observed that Lion’s conditions on the sequence {λn} excluded the canonical choice λn=1/n. This was overcome in 2003 by Xu and Kim [6] in a Hilbert space. They proved that if {λn} satisfies conditions (A1), (A2) and (A4)

(A4)limn→∞λn/λn+1=1or, equivalently,limn→∞(λn−λn+1)/λn+1=0,

then {xn} is strongly convergent to the unique solution u∗ of the variational inequality 〈Fu∗,v−u∗〉≥0, ∀v∈C. It is easy to see that condition (A4) is strictly weaker than condition (A3), coupled with conditions (A1) and (A2). Moreover, (A4) includes the important and natural choice {1/n} of {λn}.

In 2010, Tian [7] improved Yamada’s method (1.2) and established the following strong convergence theorems.

We remind the reader of the following facts: (i) The results are obtained when the underlying space is a Hilbert space in Yamada [4], Xu [6] and Tian [7]. (ii) In order to guarantee the strong convergence of the iterative sequence {xn}, there is at least one parameter sequence converging to zero (i.e., αn→∞ or λn→0) in Yamada [4], Xu [6] and Tian [7]. (iii) The parameter sequence satisfies the condition limn→∞λn/λn+1=1 (or limn→∞αn+1/αn=1).

In this paper, we establish a necessary and sufficient condition for the strong convergence of {xn} generated by (3.7) (defined below) in a uniformly convex and 2-uniformly smooth Banach space. In the meantime, we remove the control condition (C1) and replace condition (C3) with (C3′) (defined below) in the result of Tian [7]. It is worth pointing out that we use a new method to prove our main results. The results presented in this paper can be viewed as an improvement, supplement and extension of the results obtained in [4, 6, 7].

2 Preliminaries

For the sequence {xn} in X, we write xn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x means that {xn} converges strongly to x. In order to prove our main results, we need the following lemmas.

Lemma 2.5LetXbe a real 2-uniformly smooth Banach space. Lettbe a number in(0,1), and letμ>0. LetF¯:X→Xbe an operator such that, for some constant0<η≤2kK, F¯isk-Lipschitzian andη-strongly accretive. ThenS=(I−tμF¯):X→Xis a contraction providedμ≤η/(2k2K2), that is,

3 Main results

Throughout this section, let X be a uniformly convex and 2-uniformly smooth Banach space. Let T:X→X be a nonexpansive mapping with Fix(T)≠∅. Let F¯:X→X be a k-Lipschitzian and η-strongly accretive operator with 0<η≤2kK. Let μ∈(0,η/(2k2K2)] and τ=1−1−2μη+2μ2k2K2. Let f¯:X→X be a Lipschitzian and strongly pseudo-contractive operator with 0<ρ<τ. Let t be a number in (0,1). Consider a mapping St on X defined by

Stx=tf¯(x)+(I−tμF¯)Tx,x∈X.

It is easy to see that the mapping St is strongly pseudo-contractive. Indeed, from Lemma 2.5, we have

Proof It is easy to see the uniqueness of a solution of variational inequality (3.2). By Lemma 2.4, μF¯−f¯ is strongly accretive, so variational inequality (3.2) has only one solution. Below we use x∗∈Fix(T) to denote the unique solution of (3.2).

Next, we prove that {xt} is bounded. Take u∈Fix(T), from (3.1) and using Lemma 2.5, we have

Since {xt} is bounded, without loss of generality, we may assume that {xn} converges weakly to a point x˜. By (3.4) and using Lemma 2.2, we have x˜∈Fix(T). Then by (3.5), xn→x˜. This has proved the relative norm compactness of the net {xt} as t→0+.

We next show that x˜ solves variational inequality (3.2). Observe that

(μF¯−f¯)(xt)=−1t(I−T)xt+μ(F¯xt−F¯Txt).

Since I−T is accretive (this is due to the nonexpansiveness of T), for any u∈Fix(T), we can deduce immediately that

The framework of a Hilbert space is extended to a uniformly convex and 2-uniformly smooth Banach space.

(ii)

The η-strongly monotone operator F is extended to the case of an η-strongly accretive operator F¯. The contraction f:H→H is extended to the case of a Lipschitzian and strongly pseudo-contractive operator f¯:X→X.

(iii)

If we put X=H, F¯=F and f¯=γf, then our Theorem 3.1 reduces to Theorem 3.1 of Tian [7]. Thus, our Theorem 3.1 covers Theorem 3.1 of Tian [7] as a special case.

Next we consider the following iteration process: the initial guess x0 is selected in X arbitrarily and the (n+1)th iterate xn+1 is defined by

xn+1=(I−αnμF¯)Txn+αnγf(xn),n≥0,

(3.7)

where f:X→X is a contractive mapping with 0<γα<τ, {αn} is a sequence in (0,1) satisfying conditions (C2) and

(C3′) |αn+1−αn|≤o(αn+1)+σn with σn≥0 and ∑n=0∞σn<∞.

Besides the basic condition (C2) on the sequence αn, we have the control condition (C3′). It can obviously be replaced by one of the following:

(C3-1) ∑n=0∞|αn+1−αn|<∞;

(C3-2) limn→∞αn+1/αn=1.

Indeed, (C3-1) implies (C3′) by choosing σn=|αn+1−αn|, and (C3-2) implies (C3′) by choosing σn=0. In this sense (C3′) is a weaker condition than the previous condition (C3).

Our second main result below shows that we have established a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space.

Theorem 3.3Let{xn}be generated by algorithm (3.7) with the sequenceαnof parameters satisfying conditions (C2) and (C3′). Then

where M=max{γf(xn−1),μF¯Txn−1}. By Lemma 2.3, we have limn→∞∥xn+1−xn∥=0.

Step 3. We claim that limn→∞∥xn−Txn∥=0. Indeed, from Step 2, we have

∥xn−Txn∥≤∥xn−xn+1∥+∥xn+1−Txn∥=∥xn−xn+1∥+∥αn(γf(xn)−μF¯Txn)∥→0(n→∞).

Step 4. We claim that lim supn→∞〈(γf−μF¯)x∗,j(xn+1−x∗)〉≤0, where x∗=limt→0+xt and xt is defined by xt=tγf(xt)+(I−tμF¯)Txt. Since {xn+1} is bounded, there exists a subsequence {x{n+1}k} of {xn+1} which converges weakly to ω. From Step 3, we obtain Tx{n+1}k⇀ω. From Lemma 2.2, we have ω∈Fix(T). Since f is a contractive mapping, we have that γf is a Lipschitzian and strongly pseudo-contractive operator with γα∈(0,τ). Hence, using Theorem 3.1, we have x∗∈Fix(T) and

where μn=αn(τ−γα) and δn=2τ−γα〈γf(x∗)−μF¯x∗,j(xn+1−x∗)〉. It is easy to see that ∑n=1∞μn=∞ and lim supn→∞δn≤0. Hence, by Lemma 2.3, the sequence {xn} converges strongly to x∗∈Fix(T). From x∗=limt→0xt and Theorem 3.1, we have that x∗ is the unique solution of the variational inequality 〈(μF¯−γf)x∗,j(x∗−u)〉≤0, u∈Fix(T).

On the other hand, suppose that xn→x∗∈Fix(T) as n→∞, where x∗ is the unique solution of the variational inequality 〈(μF¯−γf)x∗,j(x∗−u)〉≤0, u∈Fix(T). Observe that

The framework of a Hilbert space is extended to a uniformly convex and 2-uniformly smooth Banach space.

(ii)

The η-strongly monotone operator F is extended to the case of an η-strongly accretive operator F¯.

(iii)

We establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings. It follows from (C1) that αn(γf(xn)−μF¯Txn)→0 (n→∞). Hence, we can obtain Theorem 3.2 of Tian [7] immediately. Thus, our Theorem 3.3 covers Theorem 3.1 of Tian [7] as a special case.

The following example shows that all the conditions of Theorem 3.3 are satisfied. However, condition (C1) is not satisfied.

Example 3.5 Let X=R be the set of real numbers. Define the mappings T:X→X, f:X→X and F¯:X→X as follows:

Tx=0,F¯x=xandf(x)=12x∀x∈R.

It is easy to see that K=22, α=12 and Fix(T)={0}. By F¯x=x, we have η=k=1 and hence 0<μ≤η/(2k2K2)=1. Also, put μ=1. It is easy to see that τ=1−1−2μη+2μ2k2K2=1. From 0<γα<τ, we have γ∈(0,2). Without loss of generality, we put γ=1. Given sequences {αn} and {σn}, αn=1/2, o(αn+1)=1/n2 and σn=0 for all n≥0. For an arbitrary x0∈X, let {xn} be defined as

xn+1=(I−αnμF¯)Txn+αnγf(xn),n≥0,

that is,

xn+1=12⋅12xn=14xn,n≥0.

Observe that for all n≥0,

∥xn+1−0∥=14∥xn−0∥.

Hence we have ∥xn+1−0∥=(14)n+1∥x0−0∥ for all n≥0. This implies that {xn} converges strongly to 0∈Fix(T).

Observe that 〈(μF¯−γf)0,j(0−u)〉≤0, u∈Fix(T), that is, 0 is the solution of the variational inequality 〈(μF¯−γf)x∗,j(x∗−u)〉≤0, u∈Fix(T).

Finally, we have

∥αn(γf(xn)−μF¯Txn)∥=∥12(12xn−0)∥=14∥xn∥→0(n→∞).

Hence there is no doubt that all the conditions of Theorem 3.3 are satisfied. Since αn=1/2, condition (C1): limn→∞αn=0 of Tian [7] is not satisfied.

Declarations

Acknowledgements

The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper. This study was supported by the Natural Science Foundation of Jiangsu Province under Grant (13KJB110028), and the National Science Foundation of China (11271277).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly to this research article. Both authors read and approved the final manuscript.

Copyright

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